The full specification of a superposition state |ψi = α|0i+ β|1i is given by
the complex numbers α and β. The meaning of these numbers is physically
derived by making measurements on this state, in the computational basis.
This process would randomly “collapse” the state to either |0i or |1i. The
probability of obtaining |0i is |α|
2
and of obtaining |1i is |β|
2
. This is true in
a statistical sense: make the same measurements on a statistically large set
of identically prepared qubits: an ensemble. A measurement on a single qubit
state that is unknown projects it on to a basis state and the original state is
destroyed.
So if we are given a single quantum system in the state |ψi then can
we make clones (that is, exact copies) of the state so that we can gather
the requisite measurement data? The answer given by quantum mechanics is
“NO”.
There exists no quantum mechanical way (i.e., a unitary operator) to take
one unknown state and make multiple identical copies of it.
This is the no cloning theorem first formulated in 1982 [76, 27], which
states that an arbitrary quantum system cannot be cloned by a universal
unitary transformation. If
ˆ
U
cl
is a unitary cloning machine, then its action
would be defined as taking as input the state |ψi to be cloned along with a
“blank” state, say |0i, and produce as output the original state and its clone:
ˆ
U
cl
|ψi|0i = |ψi|ψi. (4.3)
66 Introduction to Quantum Physics and Information Processing
Theorem: A unitary transformation cannot make identical copies of an
arbitrary quantum state.
Proof. Suppose there does exist a cloning machine as defined by Equation 4.3.
Consider its action on two arbitrary quantum states |ψi and |φi:
ˆ
U
cl
|ψi|0i = |ψi|ψi, (4.4a)
ˆ
U
cl
|φi|0i = |φi|φi. (4.4b)
Take the inner product of (4.4a) with (4.4b),
LHS = hφ|h0|
ˆ
U
†
cl
ˆ
U
cl
|ψi|0i
= hφ|ψi,
RHS = hφ|hφ|ψi|ψi
= hφ|ψi
2
.
The only way LHS = RHS is if hφ|ψi = 0 (they are orthogonal) or if
hφ|ψi = 1 (they are identical). Thus a more rigorous statement of the no-
cloning theorem would be that non-orthogonal states cannot be cloned by the
same unitary operator.
Another proof is as follows:
Proof. Since
ˆ
U
cl
is linear, its operation on a linear combination of states will
be
ˆ
U
cl
(|ψi + |φi)|0i = |ψi|ψi + |φi|φi.
However, a cloner of the state |ψi + |φi must produce
(|ψi + |φi)(|ψi + |φi) = |ψi|ψi + 2|ψi|φi + |φi|φi,
which is NOT what
ˆ
U
cl
produced! In fact, the output of the cloner is actually
an ENTANGLED state (Section 4.4) while what we require is a product state.
Due to this inconsistency,
ˆ
U
cl
does not exist.
You will see an illustration of this using CNOT operations in Chapter 7.
The converse of this theorem is also true. Sometimes referred to as the no
deletion theorem [52], this states that given multiple copies of an unknown
quantum state, no unitary transformation can delete one of the copies to
give a blank (|0i). This theorem thus protects the information content in a
qubit. Both these theorems are of great importance in the theory of quantum
information
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